| L(s) = 1 | + 4.88·2-s + 15.9·4-s + 7·7-s + 38.6·8-s − 54.9·11-s − 49.7·13-s + 34.2·14-s + 61.7·16-s − 133.·17-s + 138.·19-s − 268.·22-s − 7.32·23-s − 243.·26-s + 111.·28-s − 87.2·29-s − 209.·31-s − 7.32·32-s − 653.·34-s − 67.9·37-s + 679.·38-s − 77.6·41-s − 197.·43-s − 873.·44-s − 35.8·46-s − 4.97·47-s + 49·49-s − 791.·52-s + ⋯ |
| L(s) = 1 | + 1.72·2-s + 1.98·4-s + 0.377·7-s + 1.70·8-s − 1.50·11-s − 1.06·13-s + 0.653·14-s + 0.964·16-s − 1.90·17-s + 1.67·19-s − 2.60·22-s − 0.0664·23-s − 1.83·26-s + 0.751·28-s − 0.558·29-s − 1.21·31-s − 0.0404·32-s − 3.29·34-s − 0.301·37-s + 2.90·38-s − 0.295·41-s − 0.701·43-s − 2.99·44-s − 0.114·46-s − 0.0154·47-s + 0.142·49-s − 2.11·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 2 | \( 1 - 4.88T + 8T^{2} \) |
| 11 | \( 1 + 54.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 49.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 133.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 138.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 7.32T + 1.21e4T^{2} \) |
| 29 | \( 1 + 87.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 209.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 67.9T + 5.06e4T^{2} \) |
| 41 | \( 1 + 77.6T + 6.89e4T^{2} \) |
| 43 | \( 1 + 197.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 4.97T + 1.03e5T^{2} \) |
| 53 | \( 1 - 53.0T + 1.48e5T^{2} \) |
| 59 | \( 1 - 683.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 26.8T + 2.26e5T^{2} \) |
| 67 | \( 1 - 149.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 6.15T + 3.57e5T^{2} \) |
| 73 | \( 1 - 294.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 938.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 784.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 275.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.16e3T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.502118688456068605115205763311, −7.35855759697010683233702741626, −7.08589290415182441108602225381, −5.82706075926159986731728366651, −5.14268720095678151213103173917, −4.71293087374399219683751719281, −3.61802145633736243210805191720, −2.65475116913557880394848752099, −1.98888393090777271007683300240, 0,
1.98888393090777271007683300240, 2.65475116913557880394848752099, 3.61802145633736243210805191720, 4.71293087374399219683751719281, 5.14268720095678151213103173917, 5.82706075926159986731728366651, 7.08589290415182441108602225381, 7.35855759697010683233702741626, 8.502118688456068605115205763311
